\(\int \frac {1}{144} e^{\frac {1}{36} (17 e^{x/4}-17 x+e^x (-9 e^{x/4}+9 x))} (-68+17 e^{x/4}+e^x (36-45 e^{x/4}+36 x)) \, dx\) [1427]

Optimal result

Integrand size = 69, antiderivative size = 24 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{\frac {1}{4} \left (-\frac {17}{9}+e^x\right ) \left (-e^{x/4}+x\right )} \] Output:

exp(1/4*(x-exp(1/4*x))*(exp(x)-17/9))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{-\frac {1}{36} \left (-17+9 e^x\right ) \left (e^{x/4}-x\right )} \] Input:

Integrate[(E^((17*E^(x/4) - 17*x + E^x*(-9*E^(x/4) + 9*x))/36)*(-68 + 17*E 
^(x/4) + E^x*(36 - 45*E^(x/4) + 36*x)))/144,x]
 

Output:

E^(-1/36*((-17 + 9*E^x)*(E^(x/4) - x)))
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{144} \left (e^x \left (36 x-45 e^{x/4}+36\right )+17 e^{x/4}-68\right ) \exp \left (\frac {1}{36} \left (-17 x+17 e^{x/4}+e^x \left (9 x-9 e^{x/4}\right )\right )\right ) \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{144} \int -\exp \left (\frac {1}{36} \left (-9 e^x \left (e^{x/4}-x\right )+17 e^{x/4}-17 x\right )\right ) \left (-9 e^x \left (4 x-5 e^{x/4}+4\right )-17 e^{x/4}+68\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{144} \int \exp \left (\frac {1}{36} \left (-9 e^x \left (e^{x/4}-x\right )+17 e^{x/4}-17 x\right )\right ) \left (-9 e^x \left (4 x-5 e^{x/4}+4\right )-17 e^{x/4}+68\right )dx\)

\(\Big \downarrow \) 7281

\(\displaystyle -\frac {1}{36} \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )} \left (-9 e^x \left (4 x-5 e^{x/4}+4\right )-17 e^{x/4}+68\right )d\frac {x}{4}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{36} \int \left (9 e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+x} \left (-4 x+5 e^{x/4}-4\right )+68 e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )}-17 e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+\frac {x}{4}}\right )d\frac {x}{4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{36} \left (-68 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )}d\frac {x}{4}+17 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+\frac {x}{4}}d\frac {x}{4}+36 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+x}d\frac {x}{4}-45 \int e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+\frac {5 x}{4}}d\frac {x}{4}+144 \int \frac {1}{4} e^{\frac {1}{36} \left (17-9 e^x\right ) \left (e^{x/4}-x\right )+x} xd\frac {x}{4}\right )\)

Input:

Int[(E^((17*E^(x/4) - 17*x + E^x*(-9*E^(x/4) + 9*x))/36)*(-68 + 17*E^(x/4) 
 + E^x*(36 - 45*E^(x/4) + 36*x)))/144,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
risch	\({\mathrm e}^{-\frac {{\mathrm e}^{\frac {5 x}{4}}}{4}+\frac {{\mathrm e}^{x} x}{4}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\)	\(23\)
parallelrisch	\({\mathrm e}^{\frac {\left (-9 \,{\mathrm e}^{\frac {x}{4}}+9 x \right ) {\mathrm e}^{x}}{36}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\)	\(26\)
norman	\({\mathrm e}^{\frac {\left (-9 \,{\mathrm e}^{\frac {x}{4}}+9 x \right ) {\mathrm e}^{x}}{36}+\frac {17 \,{\mathrm e}^{\frac {x}{4}}}{36}-\frac {17 x}{36}}\)	\(30\)

Input:

int(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9* 
exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x,method=_RETURNVERBOSE)
 

Output:

exp(-1/4*exp(5/4*x)+1/4*exp(x)*x+17/36*exp(1/4*x)-17/36*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{\left (\frac {1}{4} \, x e^{x} - \frac {17}{36} \, x - \frac {1}{4} \, e^{\left (\frac {5}{4} \, x\right )} + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \] Input:

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 
6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x, algorithm="frica 
s")
 

Output:

e^(1/4*x*e^x - 17/36*x - 1/4*e^(5/4*x) + 17/36*e^(1/4*x))
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{- \frac {17 x}{36} + \left (\frac {x}{4} - \frac {e^{\frac {x}{4}}}{4}\right ) e^{x} + \frac {17 e^{\frac {x}{4}}}{36}} \] Input:

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 
6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x)
 

Output:

exp(-17*x/36 + (x/4 - exp(x/4)/4)*exp(x) + 17*exp(x/4)/36)
 

Maxima [F]

\[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=\int { \frac {1}{144} \, {\left (9 \, {\left (4 \, x - 5 \, e^{\left (\frac {1}{4} \, x\right )} + 4\right )} e^{x} + 17 \, e^{\left (\frac {1}{4} \, x\right )} - 68\right )} e^{\left (\frac {1}{4} \, {\left (x - e^{\left (\frac {1}{4} \, x\right )}\right )} e^{x} - \frac {17}{36} \, x + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \,d x } \] Input:

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 
6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x, algorithm="maxim 
a")
 

Output:

1/144*integrate((9*(4*x - 5*e^(1/4*x) + 4)*e^x + 17*e^(1/4*x) - 68)*e^(1/4 
*(x - e^(1/4*x))*e^x - 17/36*x + 17/36*e^(1/4*x)), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=e^{\left (\frac {1}{4} \, x e^{x} - \frac {17}{36} \, x - \frac {1}{4} \, e^{\left (\frac {5}{4} \, x\right )} + \frac {17}{36} \, e^{\left (\frac {1}{4} \, x\right )}\right )} \] Input:

integrate(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/3 
6*(-9*exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x, algorithm="giac" 
)
 

Output:

e^(1/4*x*e^x - 17/36*x - 1/4*e^(5/4*x) + 17/36*e^(1/4*x))
 

Mupad [B] (verification not implemented)

Time = 3.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx={\mathrm {e}}^{\frac {17\,{\mathrm {e}}^{x/4}}{36}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{4}}\,{\mathrm {e}}^{-\frac {17\,x}{36}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{x/4}\,{\mathrm {e}}^x}{4}} \] Input:

int((exp((17*exp(x/4))/36 - (17*x)/36 + (exp(x)*(9*x - 9*exp(x/4)))/36)*(1 
7*exp(x/4) + exp(x)*(36*x - 45*exp(x/4) + 36) - 68))/144,x)
 

Output:

exp((17*exp(x/4))/36)*exp((x*exp(x))/4)*exp(-(17*x)/36)*exp(-(exp(x/4)*exp 
(x))/4)
 

Reduce [F]

\[ \int \frac {1}{144} e^{\frac {1}{36} \left (17 e^{x/4}-17 x+e^x \left (-9 e^{x/4}+9 x\right )\right )} \left (-68+17 e^{x/4}+e^x \left (36-45 e^{x/4}+36 x\right )\right ) \, dx=-\frac {5 \left (\int \frac {e^{\frac {17 e^{\frac {x}{4}}}{36}+\frac {e^{x} x}{4}+\frac {5 x}{4}}}{e^{\frac {e^{\frac {5 x}{4}}}{4}+\frac {17 x}{36}}}d x \right )}{16}+\frac {\left (\int \frac {e^{\frac {17 e^{\frac {x}{4}}}{36}+\frac {e^{x} x}{4}+x}}{e^{\frac {e^{\frac {5 x}{4}}}{4}+\frac {17 x}{36}}}d x \right )}{4}+\frac {17 \left (\int \frac {e^{\frac {17 e^{\frac {x}{4}}}{36}+\frac {e^{x} x}{4}+\frac {x}{4}}}{e^{\frac {e^{\frac {5 x}{4}}}{4}+\frac {17 x}{36}}}d x \right )}{144}-\frac {17 \left (\int \frac {e^{\frac {17 e^{\frac {x}{4}}}{36}+\frac {e^{x} x}{4}}}{e^{\frac {e^{\frac {5 x}{4}}}{4}+\frac {17 x}{36}}}d x \right )}{36}+\frac {\left (\int \frac {e^{\frac {17 e^{\frac {x}{4}}}{36}+\frac {e^{x} x}{4}+x} x}{e^{\frac {e^{\frac {5 x}{4}}}{4}+\frac {17 x}{36}}}d x \right )}{4} \] Input:

int(1/144*((-45*exp(1/4*x)+36*x+36)*exp(x)+17*exp(1/4*x)-68)*exp(1/36*(-9* 
exp(1/4*x)+9*x)*exp(x)+17/36*exp(1/4*x)-17/36*x),x)
 

Output:

( - 45*int(e**((17*e**(x/4) + 9*e**x*x + 45*x)/36)/e**((9*e**((5*x)/4) + 1 
7*x)/36),x) + 36*int(e**((17*e**(x/4) + 9*e**x*x + 36*x)/36)/e**((9*e**((5 
*x)/4) + 17*x)/36),x) + 17*int(e**((17*e**(x/4) + 9*e**x*x + 9*x)/36)/e**( 
(9*e**((5*x)/4) + 17*x)/36),x) - 68*int(e**((17*e**(x/4) + 9*e**x*x)/36)/e 
**((9*e**((5*x)/4) + 17*x)/36),x) + 36*int((e**((17*e**(x/4) + 9*e**x*x + 
36*x)/36)*x)/e**((9*e**((5*x)/4) + 17*x)/36),x))/144